A. Quadratic Reciprocity Via Gauss Sums
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Rings of Integers, Gauss-jacobi Sums, and Their Applications
In this paper we shall explore the structure of the ring of algebraic integers in any quadratic extension of the field of rational numbers Q, develop the concepts of Gauss and Jacobi sums, and apply the theory of algebraic integers and that of Gauss-Jacobi sums to solving problems involving power congruences and power sums as well as to proving the quadratic and cubic reciprocity laws. In parti...
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Quadratic Reciprocity is arguably the most important theorem taught in an elementary number theory course. Since Gauss’ original 1796 proof (by induction!) appeared, more than 100 different proofs have been discovered. Here I present one proof which is not particularly well-known, due to George Rousseau [2]. (The proof was rediscovered more recently by (then) high-schooler Tim Kunisky [1].) Alt...
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We adapt a method of Schur to determine the sign in the quadratic Gauss sum and derive from this, the law of quadratic reciprocity.
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In fact, [Eisenstein 1850] evaluated cubes and fourth powers of Gauss sums attached to cubic and quartic characters to prove the corresponding reciprocity laws. One essential point is the p-adic approximation of Gauss sums by [Kummer 1847], generalized in [Stickelberger 1890]. Since the rings of algebraic integers generated by third or fourth roots of unity have class number one and finitely-ma...
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تاریخ انتشار 2009